Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-808811220210622Numerical solution for a class of variable order fractional integral-differential equation with Atangana-Baleanu-Caputo fractional derivativeNumerical solution for a class of variable order fractional integral-differential equation with Atangana-Baleanu-Caputo fractional derivative2532701677010.22055/jamm.2021.35430.1866FAHajimohammadMohammadinejadDepartment of Mathematics, Faculty of Science, University of Birjand, Birjand, Iran.HassanKhosraviDepartment of Mathematics, Faculty of Science, University of Birjand, Birjand, Iran.Journal Article20201014In this paper we consider fractional integral-differential equations of variable order containing Atangana-Baleanu-Caputo fractional derivatives as follows: <br />\begin{align*} <br />\mathfrak{D}_{\alpha(t)}^{ABC}&\Big[u(x,t).g(x,t)\Big]+\frac{\partial u(x,t)}{\partial t}+\int_{0}^{t}u(x,Y)dY\nonumber\\ <br />&+\int_{0}^{t}u(x,Y).k(x,Y)dY=f(x,t), <br />\end{align*} <br />We try to solve this equation using a numerical method based on matrix operators including Chebyshev polynomials. By using these operational matrixes the fractional order integral-differential equation is transformed into an algebraic system which by solving them, we will obtain the numerical answer of the above fractional integral-differential equation. To show the accuracy and efficiency of this method, we have calculated some numerical examples by MATLAB software.In this paper we consider fractional integral-differential equations of variable order containing Atangana-Baleanu-Caputo fractional derivatives as follows: <br />\begin{align*} <br />\mathfrak{D}_{\alpha(t)}^{ABC}&\Big[u(x,t).g(x,t)\Big]+\frac{\partial u(x,t)}{\partial t}+\int_{0}^{t}u(x,Y)dY\nonumber\\ <br />&+\int_{0}^{t}u(x,Y).k(x,Y)dY=f(x,t), <br />\end{align*} <br />We try to solve this equation using a numerical method based on matrix operators including Chebyshev polynomials. By using these operational matrixes the fractional order integral-differential equation is transformed into an algebraic system which by solving them, we will obtain the numerical answer of the above fractional integral-differential equation. To show the accuracy and efficiency of this method, we have calculated some numerical examples by MATLAB software.https://jamm.scu.ac.ir/article_16770_e5f0394a0f53e68e822dbf4f66207d02.pdf